{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

1Ec170NewCournotHand(2)

# 1Ec170NewCournotHand(2) - Notes on the Cournot Model I...

This preview shows pages 1–2. Sign up to view the full content.

Notes on the Cournot Model I. Assumptions: 2 identical firms, linear demand, and constant marginal cost Market Demand is P = a-bQ (where Q is market output) and MC = c. 1.Firm one’s profit function: π 1 = (a-bQ 1 -bQ 2 )(Q 1 ) – cQ 1 = (a-c)Q 1 – bQ 1 2 -bQ 2 Q 1 . 2. d π 1 /dQ 1 = (a-c)-2bQ 1 -bQ 2 = 0. Q 1 = (a-c)/2b -0.5Q 2 . Reaction Function of Firm 1. Q 2 = (a-c)/2b -0.5Q 1 . Reaction Function of Firm 2. 3. Substituting Firm 2’s reaction function into Firm 1’s reaction function & solving for Q 1 : Q 1 = (a-c)/2b -0.5[(a-c)/2b -0.5Q 1 ] = (a-c)/4b + 0.25Q 1 . 0.75Q 1 = (a-c)/4b or Q 1 = (a-c)/3b = (1/3)[(a-c)/b]. 4. Market Quantity: Q= Q 1 +Q 2 = 2(a-c)/3b = (2/3)[(a-c)/b]. 5. Price : P = a – b[2(a-c)/3b] = a- (2/3)(a-c) = (1/3)a + (2/3)c = (1/3) [a + 2c]. II. Assumptions: n identical firms, linear demand and constant marginal cost Market Demand is P = a-bQ (where Q is market output) and MC = c. Q = Q i + Q j and Q j = (n-1)Q i j=2 j=2 1.Firm i’s profit function: π i = (P-MC)(Q i ) = (a-bQ i -b Q j – c)(Q i ) = aQ i -bQ i 2 -bQ i Q j – cQ i . j=2 j=2 2. d π i /dQ i = a-c -b2Q i -b Q j = 0 . j=2 by substitution: d π 1 /dQ i = (a-c) -b2Q i -b(n-1)Q i = 0 . = (a-c) –bQ i (2+n-1) = 0. = (a-c) –bQ i (n+1) = 0 rearranging, Q i = (a-c)/[b(1+n)]. 3. Market output: Q = nQ i = [n/(1+n)][(a-c)/b]. 4. Price: P = a-bQ = a-b{[n/(1+n)][(a-c)/b]} = a - {[n/(1+n)][(a-c)]} = [1/(1+n)] [a + nc]. Notice that as n Æ , then n/(1+n) Æ 1, and so P Æ c (perfect competition outcome). And if n = 1, then P = a – ½ (a-c) (monopoly outcome). III.Assumptions: 2 non-identical firms, Market Demand: P = a- Q = a- Q 1 -Q 2 (for simplicity we assume b=1) and MC 1 = c 1 and MC 2 = c 2 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}